Neumann Laplacian in a perturbed domain

TL;DR

This paper studies how the spectrum of the Neumann Laplacian changes when a domain is slightly perturbed by removing a small measure set, showing convergence of spectra in the Hausdorff sense.

Contribution

It establishes the spectral convergence of the Neumann Laplacian under domain perturbations involving removal of measure-zero sets.

Findings

Spectrum converges in Hausdorff distance

Spectral properties are stable under small measure-zero perturbations

Results apply to domains with tiny removed subsets

Abstract

We consider a domain with a small compact set of zero Lebesgue measure of removed. Our main result concerns the spectrum of the Neumann Laplacian defined on such domain. We prove that the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.